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SageMath
E = EllipticCurve("cy1")
E.isogeny_class()
Elliptic curves in class 199920cy
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
199920.c5 | 199920cy1 | \([0, -1, 0, 2399024, 1388370880]\) | \(3168685387909439/3563732336640\) | \(-1717328059078079938560\) | \([2]\) | \(10616832\) | \(2.7610\) | \(\Gamma_0(N)\)-optimal |
199920.c4 | 199920cy2 | \([0, -1, 0, -13657296, 13090216896]\) | \(584614687782041281/184812061593600\) | \(89059140544206628454400\) | \([2, 2]\) | \(21233664\) | \(3.1076\) | |
199920.c2 | 199920cy3 | \([0, -1, 0, -198054096, 1072707988416]\) | \(1782900110862842086081/328139630024640\) | \(158127306067021297090560\) | \([2]\) | \(42467328\) | \(3.4542\) | |
199920.c3 | 199920cy4 | \([0, -1, 0, -86161616, -297866310720]\) | \(146796951366228945601/5397929064360000\) | \(2601209677794875965440000\) | \([2, 2]\) | \(42467328\) | \(3.4542\) | |
199920.c6 | 199920cy5 | \([0, -1, 0, 33790384, -1062392377920]\) | \(8854313460877886399/1016927675429790600\) | \(-490047586658875122890342400\) | \([2]\) | \(84934656\) | \(3.8008\) | |
199920.c1 | 199920cy6 | \([0, -1, 0, -1366182736, -19435718080064]\) | \(585196747116290735872321/836876053125000\) | \(403282455650726400000000\) | \([2]\) | \(84934656\) | \(3.8008\) |
Rank
sage: E.rank()
The elliptic curves in class 199920cy have rank \(1\).
Complex multiplication
The elliptic curves in class 199920cy do not have complex multiplication.Modular form 199920.2.a.cy
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.